Transient response of relativistic electron bunches to wave-number selected perturbations near the micro-bunching instability threshold

Many spatio-temporal systems can undergo instabilities, leading to the spontaneous formation of spatial structures (patterns). However, a range of cases exist for which the pattern itself is not directly visible because of technical or fundamental reasons. This is the case for the spontaneous formation of millimeter-scale patterns appearing inside relativistic electron bunches of accelerators. We demonstrate in this case how the study of responses to sine external perturbations can be used as a ‘probe’ to deduce the characteristic wavenumber of the pattern formation process. Experiments are performed in the UVSOR-II electron storage ring when the electron bunch is subjected to so-called microbunching instability, and the sine perturbations are provided by an external laser. The response is constituted of pulses of coherent synchrotron radiation, whose amplitude depends on the perturbation wavenumber. Experimental results on the dynamics are compared to numerical calculations obtained using a Vlasov–Fokker–Planck model.

S Online supplementary data available from stacks.iop.org/njp/16/063027/ mmedia Keywords: accelerators, pattern formation, Vlasov, synchrotron radiation, instabilities In a relativistic electron bunch, interactions between electrons through electric forces (wakefields) can be the cause of so-called microbunching instability [1][2][3][4][5][6][7][8][9], which is characterized by the spontaneous formation of patterns in the electron bunch phase space [10][11][12]. The understanding of this dynamics is crucial, as electron storage rings are extremely important sources for users of radiation in the VUV, x-ray, and THz ranges. However, in contrast to more classical pattern-forming systems, as in hydrodynamics, chemistry, biology, optics, etc [13], the study of pattern formation in relativistic electron bunches presents additional difficulties because of the very indirect nature of the possible measurements. This situation is illustrated in figure 1, where we have represented a typical phase space computed numerically using the parameters of the UVSOR-II storage ring. Above a threshold current, a two-dimensional pattern is spontaneously formed (figures 1(b), (c)) and its evolution in time t is usually extremely rich and complex, as the structures continuously rotate in phase space, and appear in a bursting manner. In contrast, the experimental data contain much less information because they usually consist of the recording of the coherent submillimeter/millimeter radiation (CSR) emitted by the structures using bolometers (figures 1(d), (e)), or diode detectors. In other words, the detected signal D(t) can be viewed mathematically as a projection of the twodimensional pattern δ ( ) f z e t , , onto a scalar variable D(t): 2 with H(k) a transfer function, and ( ) FF k t , the so-called form factor [14] of the electron bunch at time t. The details of this projection are not important at this stage. However, this already shows that most of the information on the two-dimensional pattern δ ( ) f z e t , , is lost, and that even crucial features such as the wavenumber selected at the instability threshold are difficult to access 5 from the detected signal D(t).
In this article, we present an alternate approach that consists of systematically examining the systemʼs transient response to initial perturbations with selected wavenumbers for various values of the main operating parameter. From common knowledge in nonlinear dynamics, one may indeed expect a huge increase in the systemʼs response when the system is near a phase transition point [13] (the threshold of the microbunching instability here), and the perturbation has a wavenumber close to the characteristic wavenumber of the instability. From a conceptual point of view, this perturbation strategy is close to the experiments performed in the Rayleigh-Bénard experiments [15], although our goal is different here.
Technically, this attempt to use laser perturbations to probe the pattern formation is also motivated by the pioneering work of Byrd, Sannibale et al [16]. They demonstrated that a huge response (a burst of intense coherent synchrotron radiation) can be obtained using short laser pulses. Here, instead of using short pulses (and thus a broadband initial perturbation), a key point is of course to shape the laser pulses with a sine profile [17] (at millimeter scale). The sinusoidally shaped laser pulses that are chosen are moreover slightly longer than the electron bunch (several centimeters) (figure 2(a)).
The experiments were conducted in the UVSOR-II storage ring, operating at an energy of = E 600 0 MeV, in single-bunch low-alpha mode [18] (the machine parameters are displayed in table 1). In these conditions, the electron bunch has a duration of 37 ps RMS at low current, a relative energy spread σ = × − E 3.4 10  (1)). The pattern rotates in phase space with a frequency of the order of 10 kHz, and its evolution is usually irregular. In contrast, the data typically available experimentally (d), (e), contain much less information, as they consist of single bolometer signals versus time. Most of the information on the two-dimensional structure, as its characteristic wavenumber, are thus not accessible. The currents for (d) and (e) are 11.1 mA and 12.9 mA, respectively. Movies of (b) and (c) can be found in the supplementary material, available at stacks. iop.org/njp/16/036027/mmedia. instability is typically of the order of 7 mA (i.e. an electron bunch charge of ≈1.2 nC). The bursts of millimeter radiation are monitored by an InSb bolometer (QMC QFI/2) with a 2 μs response time, placed on the BL6B beamline [19]. The laser pulses interact with the electron bunch in an undulator (figure 2(a), and the optical setup is similar to the one described in [17]). We used an amplified laser (Coherent Mira and Legend) delivering 3 mJ, 300 ps FWHM long chirped pulses at 800 nm, and with a 1 kHz repetition rate. These pulses are sinusoidally modulated using the chirped pulse beating technique [20], which allows the modulation period to be chosen simply by adjusting the delay of a Michelson interferometer. The spectroscopic wavenumber ν m of the perturbation can be typically chosen from the sub-ps range to tens of picoseconds.
The data are recorded as follows. For each chosen sine perturbation wavenumber ν m , we record the transient bolometer temporal trace (which roughly corresponds to the induced pattern amplitude versus time). Furthermore, the responses versus ν m are systematically recorded versus current I because the effect is expected just below the instability threshold. The data acquisition is automated, using a Lecroy oscilloscope (WR104MXI) in sequence mode, and automatic scanning of the Michelson interferometer. At each current value, we record the responses to 200 values of ν m from 0.48 to − 4.18 cm 1 (this takes 8 s). The electron beam current is decreased at a slower rate, using the natural decay of the electron bunch charge, or additional losses when needed.
Schematically, three current zones are observed. At high current ( figure 2(d)), the laser has no clear effect, and we simply observe the usual strong bursts of millimeter CSR radiation that already occur spontaneously without the laser (and for which a pattern like the ones in figure 1(b), (c) is expected theoretically). The expected 'pattern seeding' effect is observed for currents that are near the microbunching instability threshold ( ≈ I 7 mA). The signature consists of the delayed pulse [ figure 2(b), red curve] at a time of ≈60 μs. The pattern seeding effect then decreases when the current is further decreased to very low values ( ≈ − I 4 5 mA and below), where the quasi-instantaneous response to the laser perturbation eventually remains (this peak near t = 0 does not involve the nonlinear dynamics of the system and was studied in [14,17,21]).
Furthermore, systematic analysis of this bolometer signal versus modulation wavenumber ν m reveals that the seeding is effective only in a precise domain of ν m , in the 1.5-2 cm −1 range. This effect is clearly visible in figure 2(b), where the green and red curves represent the responses to wavenumber outside and inside the adequate wavenumber range for seeding. This resonance effect appears as a characteristic feature in the colorscale representation of bolometer signals versus ν m [figure 2(c)], and this feature appears essentially near the instability threshold [figure 2(d)].
We have verified that this peak at ≈ t 60 μs is unique, using scans of ν m up to − 32 cm 1 . Moreover, the relative sizes of the delayed and quasi-immediate responses depend on current, showing that the delayed peak is affected by the effect of wakefields. This will be a main comparison point with theoretical predictions.
A classical framework for theoretical studies is the one-dimensional Vlasov-Fokker-Planck approach. The temporal evolution of the electron density θ ( ) f q p , , reads [11]: ( 2 ) E wf characterizes the interaction between electrons, and is the term responsible for the microbunching instability. It represents the average longitudinal electric field created by the whole bunch, per Ampere of currentwakefield created by one electron , and the electron bunch charge density [22].
The precise expression of E wf e 1 is a main ingredient of the modeling, and depends on the electron bunch path and detailed boundary conditions in the accelerator (e.g. the vacuum chamber). For the present test, we use the wakefield produced by an electron on a curved trajectory, between two parallel plates of height h 2 , which is one of the simplest models reproducing the experimentally observed dynamics in storage rings [22]. In MKS units, the wakefield ( ) E q wf can be expressed as [23]: where the functions w and G 2 are given by equations (3.12,5.22b) of [22], and the other parameters are given in table 1. The convolution is computed in Fourier space using an approach similar to the one proposed by Qiang [24] (see [23]). Calculation of the perturbation induced by the laser is detailed in the supplementary data, available at stacks.iop.org/njp/16/ 063027/mmedia, and The integration of equation (2) is performed using the semi-Lagrangian approach. The scheme of [25,26] is implemented using a parallel MPI strategy. The longitudinal q coordinate is divided into slices of equal size, each being associated with a computer core. The computations were performed using a mesh of 400×400 points, with a physical size (in q and p) of is finally processed by a first-order filter, with 2μs response time. We systematically integrated equation (2) for various values of the modulation wavenumber beam current. In addition, since we are interested in the effect of the wakefield on the electron bunch dynamics, we also integrated, in each case, the corresponding equation 'without wakefield' (i.e. taking I = 0). This will serve as a reference case, to which departures will be attributed to wakefield effects.
The dynamical behavior is similar to the experimental one, and the simulated bolometer response is composed of two successive peaks (figure 3(e)). As in the experiment, the delayed response around 55-60 μs is strongly enhanced when the modulation wavenumber is around 1.5 cm −1 (red curve), and is extremely weak otherwise (green curve). Moreover, we can verify that the delayed peak size is dependent on the collective effects (wakefield), by comparing the data to the reference results 'without wakefield' (i.e. I = 0 and all other parameters unchanged, black curve). The phase space evolution is represented in figures 3(a)-(d). Analyses versus the excitation wavenumber (figure 3(f)) confirms that the maximum delayed response occurs at ν ≈ 1.5 m cm −1 . Further study of wavenumber selection can be performed by plotting the height of the delayed peak versus modulation wavenumber. As expected ( figure 4(a)), a good agreement between theory and experiment is found. Furthermore, it is instructive to compare these  the phase space at different times after the laser perturbation: θ = 0, π 2, π, respectively (for clarity, the initial phase space distribution (a) has been subtracted), (e) the deduced bolometer signals, and (f) the bolometer signal versus excitation wavenumber. Note that θ π = 2 corresponds to one synchrotron period (110 μs).
curves with the CSR spectrum of the 'free-running system' (i.e. without laser seeding) above the instability threshold. The emitted spectrum ( figure 4(b)) is found to coincide with the tuning curve of figure 4(a), and this agreement was found for all currents in this zone. This shows that the seeding strategy is a promising technique for examining the characteristic wavenumbers of the instability.
As this technique provides new possibilities for testing theories, we also tried to find possible differences between the theoretical and experimental findings. One difference is the ratio between the delayed response and the immediate one (the ratio is larger experimentally, as can be seen in the red curves of figures 2(b) and 3(e)). Further refinement of the theory is beyond the scope of this paper. However, we believe that this type of experiment may also point out differences between the observations and existing models in other storage rings, and help progress in refining existing theories (e.g. wakefields).
In conclusion, the manipulation of a relativistic electron bunch using specially shaped laser pulses allows us to obtain dynamical information on the spatio-temporal instability occurring in relativistic electron bunches circulating in a storage ring. This information includes the characteristic wavenumber and the shape of the transients occurring after perturbation. The present test at UVSOR-II reveals a satisfying agreement with the Vlasov-Fokker-Planck equation with a parallel-plates wakefield. Tests in other accelerators are possible a priori when a laser perturbation is achievable, in particular in storage rings equipped with a laser slicing setup.